## Info

Equation (318) describes the Hamiltonian of the harmonic oscillator with the well-known eigenenergies

labeled by the radial (ncm = 0, 1,2,...) and azimuthal (mcm = 0, ±1, ±2, ±3,...) quantum numbers. The problem is reduced to obtaining eigenenergies sn m of the relative motion Hamiltonian. The energy states of the total Hamil-tonian are labeled by CM and relative quantum numbers, lncmmcm; nrm). The coexistence of the electron-electron and the oscillator terms makes the exact analytic solution with the present special functions not possible.

4.4.2. Shifted 1/N Expansion Method

The shifted 1/N expansion method, N being the spatial dimensions, is a pseudo-perturbative technique in the sense that it proposes a perturbation parameter that is not directly related to the coupling constant [181-184]. The aspect of this method has been clearly stated by Imbo et al. [181-183] who had displayed step-by-step calculations relevant to this method. Following their work, here we only present the analytic expressions that are required to determine the energy states.

The method starts by writing the radial Schrodinger equation, for an arbitrary cylindrically symmetric potential, in a N-dimensional space as d2 (k - 1)(k - 3)

In order to get useful results from 1/k expansion, where k = k — a and a is a suitable shift parameter, the large k-limit of the potential must be suitably defined [185]. Since the angular momentum barrier term behaves like k2 at large k, so the potential should behave similarly. This will give rise to an effective _potential, which does not vary with k at large values of k resulting in a sensible zeroth-order classical result. Hence, Eq. (321) in terms of the shift parameter becomes where d2 k2[1 - (1 - a)/k][1 - (3 - a)/k] dr2 4r2

and Q is a scaling constant to be specified from Eq. (325). The shifted 1/N expansion method consists of solving Eq. (322) systematically in terms of the expansion parameter 1 /k. The leading contribution term to the energy comes from k 2 Veff (r)

where r0 is the minimum of the effective potential, given by 2r<3 V = Q (325)

It is convenient to shift the origin to r0 by the definition

and expanding Eq. (322) about x = 0 in powers of x. Comparing the coefficients of powers of x in the series with the corresponding ones of the same order in the Schrodinger equation for one-dimensional anharmonic oscillator, we determine the anharmonic oscillator frequency, the energy eigenvalue, and the scaling constant in terms of k, Q, r0 and the potential derivatives. The anharmonic frequency parameter is a =

and the energy eigenvalues in powers of 1/k (up to third order) read

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